IntegralSam

Hey! I'm not very interesting ...

Integration Trick I've Accumulated!

I'll probably add more onto here sometime later, but I think this is a good start!

$\int _{-\infty }^{\infty }e^{-ax^{2}+bx+c}dx=e^{c+{\frac {b^{2}}{4a}}}{\sqrt {\frac {\pi }{a}}}$

$\int _{0}^{\infty }{\frac {ln(x)}{1+x^{2}}}dx=0$

$\int _{0}^{\infty }{\frac {1}{(1+x^{2})(1+x^{n})}}dx={\frac {\pi }{4}},n>0$

$\int _{0}^{1}{\frac {x^{n}-1}{ln(x)}}dx=ln(x+1),n>-1$

$\int _{0}^{\infty }f(x)dx={\frac {1}{2}}\int _{0}^{\infty }\left(f(x)+{\frac {f({\frac {1}{x}})}{x^{2}}}\right)dx$

$\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx$

Here are more common tricks

• Tangent half-angle substitution
• Feynman's Integration Trick/ Differentiation Under the Integral Sign
• Odd and even functions
  • $\int _{-a}^{a}{\frac {sin(x)}{1+x^{100}}}dx=0$ . The bounds are symmetrical, thus the magnitude of the areas will cancel each other out.

This user is interested in
Mathematics.